Classification of compact homogeneous spaces with invariant $G_2$-structures

TL;DR

This paper classifies all compact homogeneous spaces that admit invariant G2-structures, revealing new examples and families of such structures, including those with nontrivial fundamental groups.

Contribution

It provides a complete classification of homogeneous spaces with invariant G2-structures under compact Lie groups, including new examples and rigidity analyses.

Findings

Many new examples with nontrivial fundamental group.

Existence of families of invariant coclosed G2-structures.

Classification includes spaces with both high and low rigidity.

Abstract

In this note we classify all homogeneous spaces $G/H$ admitting a $G$-invariant $G_2$-structure, assuming that $G$ is a compact Lie group and $G$ acts effectively on $G/H$. They include a subclass of all homogeneous spaces $G/H$ with a $G$-invariant $\tilde G_2$-structure, where $G$ is a compact Lie group. There are many new examples with nontrivial fundamental group. We study a subclass of homogeneous spaces of high rigidity and low rigidity and show that they admit families of invariant coclosed $G_2$-structures (resp. $\tilde G_2$-structures).